We will need the packages corrplot, FactoMineR and factoextra:

• Check that corrplot, FactoMineR and factoextra are installed
• If not, install them, then load them

library(corrplot)
library(FactoMineR)
library(factoextra)

We also need to load dplyr, tidyr and the “fruits” data, :

library(dplyr)
library(tidyr)
data("fruits", package = "ReMUSE")

• In the supervised methods, we observe both a set of features/variables (e.g. gene expression) for each object, as well as a response or outcome variable (e.g. metastasis information or survival information of the patients).
• The goal is then to predict the response using the variables (e.g. which genes predict best or are associated with the survival of the patients).
• Here we instead focus on exploratory analysis, we are not interested (yet) in prediction.

• Principal Component Analysis, whose goal is to extract the most “important” sources of variability in the quantitative data

• Hierachical Agglomerative Clustering, whose goal is to define clusters of samples

Covariance and correlation :

• reminders, and/or not
• visualization

Principal Component Analysis:

• principle (vocabulary!), method, a little bit of theory
• PCA with FactoMineR and visualization with factoextra

How far away is a “dot” from the barycenter ? Individual rectangle area :

\[ \left(x_i - \bar x\right) \times \left(y_i - \bar y\right) \]

The covariance is (almost) the mean area:

\[ \operatorname{cov} (x, y) = \frac{1}{n-1} \sum_{i=1}^n\left(x_i - \bar x\right) \times \left(y_i - \bar y\right) \]

Covariance can vary between \(-\infty\) and \(+\infty\).

Correlation is, by definition, a measure of linear relationship between -1 and +1:

• -1 represents a perfect negative linear relationship,
• 0 represents no linear relationship,
• +1 represents a perfect positive linear relationship.

\[ \operatorname{cor}(x, y) = \frac{\displaystyle\sum_{i = 1}^n (x_i - \bar x)(y_i - \bar y)}{\displaystyle\sqrt{\sum_{i = 1}^n (x_i - \bar x)^2}\sqrt{\sum_{i = 1}^n (y_i - \bar y)^2}} \]

… in short…

\[ \operatorname{cor}(x, y) = \frac{\operatorname{cov}(x, y)}{\operatorname{sd}(x) \operatorname{sd}(y) } \]

Compute the correlation between fruits$Potassium and fruits$Phosphore.

Constraints!

Use only the following functions/operations:

• length to compute \(n\),
• * to multiply, / to divide, - to substract,
• mean to compute the mean,
• sd to compute the standard-deviation,
• sum for the “\(\sum_{i=1}^n\)”
• as many brackets as you need

Often noted \(\rho\). Same as Pearson’s, but on the ranks! Let:

• \(r_x\) be the ranks of \(x\),
• \(r_y\) be the ranks of \(y\).

\[ \rho(x, y) = \operatorname{cor}(r_x, r_y) \]

Key properties:

• not sensitive to extreme values,
• invariant by monotonous (increasing) transformation (e.g. log, square-root),
• not adapted when there are ex-aequos

Pairs of points: select two samples \(i\) and \(j\).

• Concordant Pair : \(\left(x_{i}<x_{j} \text { et } y_{i}<y_{j}\right)\) OU \(\left(x_{i}>x_{j} \text { et } y_{i}>y_{j}\right)\)
• Discordant Pair : \(\left(x_{i}<x_{j} \text { et } y_{i}>y_{j}\right)\) OU \(\left(x_{i}>x_{j} \text { et } y_{i}<y_{j}\right)\)

\[ \tau(x, y) = \displaystyle \frac{n_C - n_D}{n_0}, \] where \(n_C\) is the number of concordant pairs, \(n_D\) the number of discordant pairs and \(n_0\) total number of pairs.

Datasaurus

Make a correlogram on the fruit data and change the color of the labels.